Continuity mistake: When Koopa is chewing out Iggy and Spike for forgetting about the rock, the amount of wax on his hands changes multiple times between cuts. For instance, the wax covering his left hand disappears in one wide shot, but is present in every other shot. Additionally, the amount of wax on his right hand changes in amount several times.
Revealing mistake: During the car chase, when Mario and Luigi have the cops on either side shooting the guns, Mario screams "Brace yourself, now!" When he says this, you can quickly see Mario and Luigi have been replaced by stunt drivers. Also, Mario's mouth isn't moving. It happens quickly, so slow motion helps, but isn't required.
Audio problem: When Mario and Luigi are escaping the police station, after they see the Bob-omb, Mario says, "Goombas! C'mon! Let's go!" and Luigi says, "Look, it was trying to give us that thing!" Their mouth movements don't quite match what they're saying. (Granted, they're facing away from camera for most of the shot, but when their mouths are on camera, you can tell they're not saying that.)
Audio problem: When Koopa and Lena are introduced, there's a long wide shot of them descending some stairs while Koopa is talking about their master plan. ("I'll be able to merge our world with theirs and get rid of the mammals!", etc.) However, Koopa's mouth blatantly isn't moving at all for most of what he says. He just has an evil grin on his face.
Revealing mistake: When Mario and Luigi first arrive in "Dinohattan" and fall through the railing, you briefly get a clear look at Luigi's stunt-double, who appears to be a bulkier man who looks nothing like John Leguizamo. It goes by quick, so it can be hard to see, but once you notice, it's obvious.
Visible crew/equipment: When Luigi and Mario are chasing after Iggy, Spike and Daisy, during the shot where Mario is talking about "breaking every bone in their body," watch the shadow at the bottom right of the screen. At one point, you can see movement in the shadow from a crew member that got caught in the light.